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Standard Deviation Practice

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Now we’re going to do the equation I showed you at the beginning, the master challenge for practicing notation, that big equation, standard deviation. Here’s this equation before and now we look at it and we see, we know what all these parts are. We’ve got an n in there, which represents number of scores. We’ve got a sigma notation with some instructions to do some subtraction and squaring, but we can do all that. And then all of this is wrapped up within one big square root, which we’re clearly going to have to do at the end because I can’t square root a sigma notation until I figure out what that number is. So let’s follow these steps one at a time.
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In general, when you see a formula like this, you want to start at the middle and work your way out. So let’s start with that. We’re going to start with this sigma notation on the top. So this sigma notation says the sigma of and then x minus the mean squared. We found the mean before. The mean is just a number. 38.75. It’s just like saying x minus 2 squared or x minus any number squared. And that’s in the parenthesis, so I’m specifically going to do that subtraction first, because I’ve got that nested in a parenthesis.
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So I’m going to take every score that x, I’m going to subtract that mean, and I’m going to do that subtraction first, and then I’m going to square those values in my sigma notation. So if we do that subtraction first, now I’ve got a negative 8.75 for about a negative 17.75, 20.25, and 6.25. So I’m doing my subtraction first because I put it in that inner parentheses. Then the instructions say to square that value. So let’s square those numbers and now I’ve got some large positive numbers. And then I’m going to execute my sigma. I can sum them all up and I’ll get 840.75.
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So as long as we follow our order of operations, this somewhat intimidating equation is actually just a simple set of instructions. Now the other thing that I know about this is the bottom said n minus 1, and in fact I can find n minus 1, because n is just the number of scores, which is 4. Because they were 4 ages. 4 minus 1 is 3. So if I plug both these things in together I get 840.75 over 3. Now I need to square root this, but of course I can’t square root a fraction unless I execute the fractions. So let’s do my division and then standard deviation is 16.74. And there you have it.
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You have defeated the most intimidating formula we’re going to learn in this course. And in fact, this is a good lesson. Any time we do a complex formula, you just have to break it down to its steps. And we haven’t learned what this thing is yet, we haven’t learned what standard deviation is, but that’s not really the point. We’ve practised with the formula notation, we now have everything we need to do pretty much any formula you’ll see in statistics or data analysis.
Now that we understand how to find the mean, let’s have a look at how to apply standard deviation.

Remember that we had a look at this equation earlier in the course:

(hat{sigma} = sqrt{frac{Sigma ((x – bar{x})^2)}{n-1}})

Up to now, we gained an understanding all of the pieces of information in this formula:

  • (n) = the number of scores
  • (x) = the data set we are working with
  • (bar{x}) = average of the data set
  • (Sigma) = the sum of the data set

We should now be able to put all of this knowledge together to calculate the Standard Deviation. Let’s have a look at this in small steps using our ages from the previous examples:

Starting with (x):
(x = 30, 21, 59, 45)

Now, let’s calculate (bar{x}):
(bar{x} = frac{30 + 21 + 59 + 45}{4})
(therefore bar{x} = 38.75)

Next, let’s look at how we can calculate (Sigma):
(Sigma ((x – bar{x})^2) = (30 – 38.75)^2 + (21 – 38.75)^2 + (59 – 38.75)^2 + (45 – 38.75)^2)
(therefore Sigma ((x – bar{x})^2) = (-8.75)^2 + (-17.75)^2 + (20.25)^2 + (6,25)^2)
(therefore Sigma ((x – bar{x})^2) = 76.5625 + 315.0625 + 410.0625 + 39.0625)
(therefore Sigma ((x – bar{x})^2) = 840.75)

And remember that we have four points of data, so:
(n = 4)

Let’s apply all of these into our formula:
(hat{sigma} = sqrt{frac{Sigma ((x – bar{x})^2)}{n-1}})
(therefore hat{sigma} = sqrt{frac{840.75}{4-1}})
(therefore hat{sigma} = sqrt{frac{840.75}{3}})
(therefore hat{sigma} = sqrt{280.25})
(therefore hat{sigma} = 16.74)

We shall discuss what standard deviation means later in this course. For now, we are learning how to apply formulas to statistics.

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